Pair Of Eigenvalues Research Articles

Bifurcations of Multi-Vortex Configurations in Rotating Bose–Einstein Condensates

We analyze global bifurcations along the family of radially symmetric vortices in the Gross–Pitaevskii equation with a symmetric harmonic potential and a chemical potential µ under the steady rotation with frequency $${\Omega}$$ . The families are constructed in the small-amplitude limit when the chemical potential µ is close to an eigenvalue of the Schro dinger operator for a quantum harmonic oscillator. We show that for $${\Omega}$$ near 0, the Hessian operator at the radially symmetric vortex of charge $${m_0 \in \mathbb{N}}$$ has m 0(m 0+1)/2 pairs of negative eigenvalues. When the parameter $${\Omega}$$ is increased, 1+m 0(m 0-1)/2 global bifurcations happen. Each bifurcation results in the disappearance of a pair of negative eigenvalues in the Hessian operator at the radially symmetric vortex. The distributions of vortices in the bifurcating families are analyzed by using symmetries of the Gross–Pitaevskii equation and the zeros of Hermite–Gauss eigenfunctions. The vortex configurations that can be found in the bifurcating families are the asymmetric vortex (m 0 = 1), the asymmetric vortex pair (m 0 = 2), and the vortex polygons $${(m_0 \geq 2)}$$ .

Morse Theory and Relative Equilibria in the Planar n-Vortex Problem

Morse theoretical ideas are applied to the study of relative equilibria in the planar $n$-vortex problem. For the case of positive circulations, we prove that the Morse index of a critical point of the Hamiltonian restricted to a level surface of the angular impulse is equal to the number of pairs of real eigenvalues of the corresponding relative equilibrium periodic solution. The Morse inequalities are then used to prove the instability of some families of relative equilibria in the four-vortex problem with two pairs of equal vorticities. We also show that, for positive circulations, relative equilibria cannot accumulate on the collision set.

Classes of non-parity-time-symmetric optical potentials with exceptional-point-free phase transitions.

Paraxial linear propagation of light in an optical waveguide with material gain and loss is governed by a Schrödinger equation with a complex potential. In this Letter, new classes of non-parity-time (PT)-symmetric complex potentials featuring conjugate-pair eigenvalue symmetry are constructed by operator symmetry methods. Due to this eigenvalue symmetry, it is shown that the spectrum of these complex potentials is often all-real. Under parameter tuning in these potentials, a phase transition can also occur, where pairs of complex eigenvalues appear in the spectrum. A distinctive feature of the phase transition here is that the complex eigenvalues may bifurcate out from an interior continuous eigenvalue inside the continuous spectrum; hence, a phase transition takes place without going through an exceptional point. In one spatial dimension, this class of non-PT-symmetric complex potentials is of the form V(x)=h'(x)-h2(x), where h(x) is an arbitrary PT-symmetric complex function. These potentials in two spatial dimensions are also derived. Diffraction patterns in these complex potentials are further examined, and unidirectional propagation behaviors are demonstrated.

Rich dynamics caused by delay in a nonchaotic Rulkov map

In this paper, we propose a simple delayed nonchaotic Rulkov map and criteria for the existence of the critical stable boundary of the unique fixed point is analyzed for $$\tau =0, 1, 2$$ , through which the equilibrium loses its stability and there occur multiple bifurcations. Compared with $$\tau =0$$ (without delay), we find that the corresponding stable region becomes larger as delay $$\tau $$ increases and interesting phenomena are discovered, including the simultaneous occurrence of two pairs of conjugate complex eigenvalues with modulus equal to 1 and $$\lambda ^n=1$$ $$(n=2,3)$$ related to strong resonance, etc. Geometrical description of the corresponding critical eigenvalue curves is also included.

Bifurcations of Heteroclinic Loop with Twisted Conditions

The bifurcation problems of twisted heteroclinic loop with two hyperbolic critical points are studied for the case [Formula: see text], [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] are the pair of principal eigenvalues of unperturbed system at the critical point [Formula: see text], [Formula: see text]. Under the transversal conditions, the authors obtained some results of the existence and the number of 1-homoclinic loop, 1-periodic orbit, double 1-periodic orbit, 2-homoclinic loop and 2-periodic orbit. Moreover, the relative bifurcation surfaces and the existence regions are given, and the corresponding bifurcation graphs are drawn.

Pseudo-Hermitian Hamiltonians generating waveguide mode evolution

We study the properties of Hamiltonians defined as the generators of transfer matrices in quasi- one-dimensional waveguides. For single- or multi-mode waveguides obeying flux conservation and time-reversal invariance, the Hamiltonians defined in this way are non-Hermitian, but satisfy symmetry properties that have previously been identified in the literature as "pseudo Hermiticity" and "anti-PT symmetry". We show how simple one-channel and two-channel models exhibit transitions between real, imaginary, and complex eigenvalue pairs.

Multi-pulse chaotic motions of functionally graded truncated conical shell under complex loads

The global bifurcations and multi-pulse orbits of an aero-thermo-elastic functionally graded material (FGM) truncated conical shell under complex loads are investigated with the case of 1:2 internal resonance and primary parametric resonance. The method of multiple scales is utilized to obtain the averaged equations. Based on the averaged equations obtained, the normal form theory is employed to find the explicit expressions of normal form associated with a double zero and a pair of pure imaginary eigenvalues. The energy-phase method developed by Haller and Wiggins is used to analyze the multi-pulse homoclinic bifurcations and chaotic dynamics of the FGM truncated conical shell. The analytical results obtained here indicate that there exist the multi-pulse Shilnikov-type homoclinic orbits for the resonant case which may result in chaos in the system. Homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are found. The diagrams show a gradual breakup of the homoclinic tree in the system as the dissipation factor is increased. Numerical simulations are presented to illustrate that for the FGM truncated conical shell, the multi-pulse Shilnikov-type chaotic motions can occur. The influence of the structural-damping, the aerodynamic-damping, and the in-plane and transverse excitations on the system dynamic behaviors is also discussed by numerical simulations. The results obtained here mean the existence of chaos in the sense of the Smale horseshoes for the FGM truncated conical shell.

Computing the closest small-signal security boundary in the control parameter space for large scale power systems

A constrained optimization algorithm is proposed for the determination of the minimum change in the parameter values of a set of controllers that locates a complex pair of eigenvalues of a linearized power system model at a user-defined small-signal security boundary. The intended practical use is to assess the combined robustness of the system controllers in maintaining adequately damped power system oscillations. The oscillation damping security margins, given an operating point and a set of fixed damping controller parameters, is the Euclidean norm of the relative parameter variation vector, also referred in the literature as the minimum distance in the control parameter space. The computational algorithm to find the Closest Security Boundary in the Control Parameter Space, CSBCPS, is based on the rigorous mathematical implementation of the non-linear programming method to the problem, including constraints on the parameter ranges. The resulting equations are solved by the Newton method. Numerical results for a large practical power system dynamic model are detailed described to illustrate the use of the proposed CSBCPS algorithm in small-signal applications involving multiple thyristor controlled series compensators and power system stabilizers.

Multipulse Orbits and Chaotic Dynamics of an Aero-Elastic FGP Plate Under Parametric and Primary Excitations

The multipulse global bifurcations and chaotic dynamics of a simply supported Functionally Graded Piezoelectric (FGP) rectangular plate with bonded piezoelectric layer are investigated with the case of 1:2 internal resonance and primary parametric resonance. Based on the averaged equations obtained, the theory of normal form is utilized to obtain the explicit expressions of normal form with a double zero and a pair of pure imaginary eigenvalues. According to the explicit expressions of normal form, the extended Melnikov method developed by Camassa et al. is employed to study the Shilnikov-type multipulse homoclinic bifurcations and chaotic dynamics of the aero-elastic FGP plate. The analytical results indicate that there exists the Shilnikov-type multipulse chaotic dynamics for the FGP plate. Numerical simulations are presented to show that for the FGP plate, the Shilnikov-type multipulse chaotic motions can occur. The influence of the in-plane excitation and the piezoelectric voltage excitation to the system dynamic behaviors is also discussed by numerical simulations. The results obtained here imply the existence of chaos in the sense of the Smale horseshoes for the FGP plate.

Localization of supersymmetric field theories on non-compact hyperbolic three-manifolds

We study supersymmetric gauge theories with an R-symmetry, defined on non-compact, hyperbolic, Riemannian three-manifolds, focusing on the case of a supersymmetry-preserving quotient of Euclidean AdS3. We compute the exact partition function in these theories, using the method of localization, thus reducing the problem to the computation of one-loop determinants around a supersymmetric locus. We evaluate the one-loop determinants employing three different techniques: an index theorem, the method of pairing of eigenvalues, and the heat kernel method. Along the way, we discuss aspects of supersymmetry in manifolds with a conformal boundary, including supersymmetric actions and boundary conditions.

Analysis of spontaneous oscillations for a three-state power-stroke model.

Our study considers the mechanism of the spontaneous oscillations of molecular motors that are driven by the power stroke principle by applying linear stability analysis around the stationary solution. By representing the coupling equation of microscopic molecular motor dynamics and mesoscopic sarcomeric dynamics by a rank-1 updated matrix system, we derived the analytical representations of the eigenmodes of the Jacobian matrix that cause the oscillation. Based on these analytical representations, we successfully derived the essential conditions for the oscillation in terms of the rate constants of the power stroke and the reversal stroke transitions of the molecular motor. Unlike the two-state model, in which the dependence of the detachment rates on the motor coordinates or the applied forces on the motors plays a key role for the oscillation, our three-state power stroke model demonstrates that the dependence of the rate constants of the power and reversal strokes on the strains in the elastic elements in the motor molecules plays a key role, where these rate constants are rationally determined from the free energy available for the power stroke, the stiffness of the elastic element in the molecular motor, and the working stroke size. By applying the experimentally confirmed values to the free energy, the stiffness, and the working stroke size, our numerical model reproduces well the experimentally observed oscillatory behavior. Furthermore, our analysis shows that two eigenmodes with real positive eigenvalues characterize the oscillatory behavior, where the eigenmode with the larger eigenvalue indicates the transient of the system of the quick sarcomeric lengthening induced by the collective reversal strokes, and the smaller eigenvalue correlates with the speed of sarcomeric shortening, which is much slower than lengthening. Applying the perturbation analyses with primal physical parameters, we find that these two real eigenvalues occur on two branches derived from a merge point of a pair of complex-conjugate eigenvalues generated by Hopf bifurcation.

Discrete‐time global leader‐following consensus of a group of general linear systems using bounded controls

SummaryThis paper deals with the problem of global leader‐following consensus of a group of discrete‐time general linear systems with bounded controls. For each follower agent in the group, we construct both a bounded state feedback control law and a bounded output feedback control law. The feedback laws for each input of an agent use a multi‐hop relay protocol, in which the agent obtains the information of other agents through multi‐hop paths in the communication network. The number of hops each agent uses to obtain its information about other agents for an input is less than or equal to the sum of the number of real eigenvalues on the unit circle and the number of pairs of complex eigenvalues on the unit circle of the subsystem corresponding to the input, and the feedback gains are constructed from the adjacency matrix of the communication network. We show that these control laws achieve global leader‐following consensus when the communication topology among follower agents forms a strongly connected and detailed balanced directed graph and the leader is a neighbor of at least one follower agent. Copyright © 2017 John Wiley & Sons, Ltd.

Orbital Normal Forms for a family of-zero Singularity

Consider a Dynamical system x'=F(x,µ) such that its linear part has a pair of imaginary eigenvalues and one zero eigenvalue (Hopf zero singularity). Recently, the simplest normal form for this singular system has been obtained by sl(2) Lie algebra theory and the decomposition of space into three invariant subspaces. The normal form of this singular system is divided into three general cases. In this paper, the obtained results will be extended to orbital normal form and one of the three aforementioned cases will be discussed. The orbital obtained normal form will be simpler than the previous simplest normal form.

Multi-hump solutions with small oscillations at infinity for stationary Swift–Hohenberg equation

The paper considers the stationary Swift–Hohenberg equationcw−(∂x2+k02)2w−w3=0,where c > 0 is a constant, , and is a small parameter. In this case, the linear operator has a pair of real eigenvalues and a pair of purely imaginary eigenvalues. It can be proved that the equation has homoclinic (or single hump) solutions approaching to periodic solutions as (called single-hump generalized homoclinic solutions). This paper provides the first rigorous proof of existence of homoclinic solutions with two humps which tend to periodic solutions at infinity (or two-hump generalized homoclinic solutions) by pasting two appropriate single-hump generalized homoclinic solutions together. The dynamical system approach is used to reformulate the problem into a classical dynamical system problem and then the solution is decomposed into a decaying part and an oscillatory part at positive infinity. By adjusting some free constants and modifying the single-hump generalized homoclinic solution near negative infinity, it is shown that the solution is reversible with respect to a point near negative infinity. Therefore, the translational invariant and reversibility properties of the system yield a two-hump generalized homoclinic solution. The method may be applied to prove the existence of 2k-hump solutions for any positive integer k.

Design of Type-2 Fuzzy Logic Power System Stabilizer Using Limit cycle approach

Power system is subjected to wide range of operating conditions. Heavily loaded power systems are subjected to Hopf bifurcations resulting in oscillatory instability. A power system shifts to the dynamic instability region encountered by unstable limit cycles, this result in increase of oscillatory behaviour of power system. Many of the authors suggested the theories like Poincare-Bendixson theory, bifurcation theory, eigenvalue theory and shooting methods for finding dynamic stability margin. Small signal stability focuses on eigenvalue computation at a given operating point. The onset of instability is obtained by finding Hopf bifurcations associated with a pair of imaginary eigenvalues using eigenvalue analysis. Computation of these HBs is a tedious and complex task in traditional approach. In this paper, a novel computational algorithm is proposed to identify the loading conditions resulting in these limit cycles, which are associated with a pair of imaginary eigenvalues, using genetic algorithm for single machine infinite bus (SMIB) and multi machine power system (MMPS) models. The derived loading conditions are used to systematically design a robust type-2 fuzzy logic power system stabiliser. The efficacy of the proposed controller is tested for linear and nonlinear models of power system under various operating conditions.

Unstable response of 2-DoF gyroscopic systems with stable eigenvalues

Gyroscopic conservative dynamical systems may exhibit flutter instability that leads to a pair of complex conjugate eigenvalues, one of which has a positive real part and thus leads to a divergent free response of the system. When dealing with non-conservative systems, the pitch fork bifurcation shifts toward the negative real part of the root locus, presenting a pair of eigenvalues with equal imaginary parts, while the real parts may or may not be negative. Several works study the stability of these systems for relevant engineering applications such as the flutter in airplane wings or suspended bridges, brake squeal, etc., and a common approach to detect the stability is the complex eigenvalue analysis that considers systems with all negative real part eigenvalues as stable systems. This paper studies the cases where the free response of these systems exhibits a transient divergent time history even if all the eigenvalues have negative real part thus usually considered as stable, and relates such a behavior to the non-orthogonality of the eigenvectors and addresses the forced response of these system, highlighting how and in which cases, an unexpected amplification of the forced response may occur.

ミスチューンを有する翼・ディスク系の振動応答に関する研究（応力で評価した高次モードのミスチューン特性）

Although bladed disks of turbomachinery are nominally designed to be cyclically symmetric (tuned system), the vibration characteristics of all blades on a disk are slightly different due to the manufacturing tolerance, the deviation of the material property, the wear during operation, and so on. These small variations break the cyclic symmetry, and split the eigenvalue pairs. The actual bladed disks with the small variations are referred to a mistuned system. Many researchers have studied mistuning, and main conclusions are while mistuning has an undesirable effect on the forced response, it has a beneficial (stabilizing) effect on the blade flutter (the self-excited vibration). Although such mistuning phenomena of bladed disks have been studied since 1980s, almost all studies focused on the amplification factor of the displacement response, and few studies researched the amplification factor of the vibratory stress response. Therefore, in the previous paper, authors studied the amplification factor expressed by the vibratory stress for the lower modes of the bladed disk, using the simple assumption. In this study, the mistuning effect expressed by the vibratory stress for the lower and higher modes are examined, using the reduced order model without any assumptions. First, formulation for evaluating the mistuning effect expressed by the vibratory stress is derived, using the reduced order model SNM (Subset of Nominal Modes). Second, the frequency response analysis of the mistuned simple bladed disk consisting of flat plates is carried out systematically. Finally, comparing the amplification factor of the displacement response with that of the vibratory stress response including the synthesized stress (Mises stress and the principal stress), mistuning phenomena expressed by the vibratory stress are clarified.

Bender-Wu singularities

We consider a family of quantum Hamiltonians $H_\hbar=-\hbar^2\,(d^2\!/dx^2) +V(x)$, $x\in\mathbb{R},$ $\hbar>0,$ where $V(x)=i(x^3-x)$ is an imaginary double well potential. We prove the existence of infinite eigenvalue crossings with the selection rules of the eigenvalue pairs taking part in a crossing. This is a semiclassical localization effect. The eigenvalues at the crossings accumulate at a critical energy for some of the Stokes lines.

Intrinsic character of Stokes matrices

Two germs of linear analytic differential systems xk+1Y′=A(x)Y with a non-resonant irregular singularity are analytically equivalent if and only if they have the same eigenvalues and equivalent collections of Stokes matrices. The Stokes matrices are the transition matrices between sectors on which the system is analytically equivalent to its formal normal form. Each sector contains exactly one separating ray for each pair of eigenvalues. A rotation in S allows supposing that R+ lies in the intersection of two sectors. Reordering of the coordinates of Y allows ordering the real parts of the eigenvalues, thus yielding triangular Stokes matrices. However, the choice of the rotation in x is not canonical. In this paper we establish how the collection of Stokes matrices depends on this rotation, and hence on a chosen order of the projection of the eigenvalues on a line through the origin.

Zero-Hopf bifurcation and multistability coexistence on a four-neuron network model with multiple delays

In this paper, a four-neuron neural system with four delays is investigated to exhibit the effects of multiple delays and coupled weights on system dynamics. Zero-Hopf bifurcation is obtained, where the system characteristic equation has a simple zero and a simple pair of pure imaginary eigenvalues. The coupled weight and time delay are considered as bifurcation parameters to study dynamical behaviors derived from zero-Hopf bifurcation. Various dynamical behaviors are analyzed near the bifurcation singularity qualitatively and quantitatively in detail by using perturbation- incremental method, and bifurcation diagrams are obtained. Numerical simulations and theoretical results are given to display a stable resting state, multistability coexistence of two resting states and a pair of periodic activities in the neighbor of the zero-Hopf bifurcation point.